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194        XIII    INTEGRATION

consequently (13.8.1) the function/^ = supfgn is A-integrable, and

r

\fgndL
Hence g is locally A-integrable (13.13.1) and g - A = sup(gn  A) (13.4.4).

n

Scholium

(13.15.3.6) We may therefore say that, for a positive measure p, the mapping
g\-*g - p is a linear bijection of the vector space Lj^^X, p) of equivalence
classes (relative to p) of locally p-integrable real-valued functions onto the
vector space of real measures with base p, and moreover that this bijection is
order-preserving. This is the fundamental fact that we shall use repeatedly
from now on until the end of Section 13.19.

We remark also that the mapping g\-*g - p, considered as a mapping into
M(X), is continuous when L^C) R(X, p) is endowed with the topology defined
in (13.13.4) and M(X) is endowed with the vague topology.

(13.15.4) In the space MR(X), every subset H which is bounded above has a
least upper bound v, and there exists an increasing sequence (jun) of elements ofH
such that v = sup

If/jjg; 0 belongs to ft R(X), then the set of numbers ju(/), where /i e H, has
a finite least upper bound by hypothesis, and we shall show that   sup \i(f} is

equal to v(/). For this purpose consider an increasing sequence (Kn) of
compact subsets of X which cover X and are such that Jf R(X) is the union
of the ^R(X; Kn) (3.18.3), and let (gmr^m^i be a dense sequence in the Banach
space JTR(X; KJ ((7.4.4) and (3.10.9)). For each pair (m, n), there exists a
sequence (^mnp)p^ of elements of H such that sup(^mnp(g^n)) = sup /*($).

p                                  n e H

Now let lr denote

sup      (nmnp)

mSjjr, n^r, p^r

(which exists by (13.15.3)). The sequence (/lr) is increasing and bounded above
in MR(X), hence has a least upper bound v0 which is also its limit in the vague
topology (13.4.4). If //e H, we have /X#mW) ^ Vo(ffJ for each pair (m,ri);
hence, by continuity and because |/+ -#*J ^ I/-&J, it follows that
X/) ^ vo(/) for all/^ 0 belonging to Jf R(X; KJ and for all /; hence ^ g v0 .
almost everywhere with respect to <r; but this implies that